Finite Group Databasebeta

← #44561861

Subgroups — #44561861

Order 32128.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548740C211yesmaximal, characteristic, complemented
2C22×#16860797C211yesmaximal, characteristic, complemented
3D4×D1004C211yesmaximal, characteristic, complemented
4#30632077C211yesmaximal, characteristic, complemented
5#30631935C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631887C211yesmaximal, characteristic, complemented
9#30631837C211yesmaximal, characteristic, complemented
10#30631832C211yesmaximal, characteristic, complemented
11#30631826C211yesmaximal, characteristic, complemented
12#30631820C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631759C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×D4×D251—21no—
19#22548740—24no—
20C2×#16860797—24no—
21C2×#16860797C2211yescharacteristic
22C2×#16860797C2211yescharacteristic
23C2×#16860797C2211yescharacteristic, complemented
24C2×#16860797C2211yescharacteristic, complemented
25C2×#16860797C2211yescharacteristic, complemented
26C2×#16860797C2211yescharacteristic, complemented
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860795C2211yescharacteristic, complemented
30C2×#16860794C2211yescharacteristic, complemented
31C22×C251⋊C8C2211yescharacteristic, complemented
32#22548712C2211yescharacteristic
33#22548690—21no—
34#22548689C2211yescharacteristic
35#22548689C2211yescharacteristic
36#22548685—21no—
37#22548685—21no—
38C4×D1004C2211yescharacteristic
39#22548648C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548636C2211yescharacteristic, complemented
44#22548635C2211yescharacteristic
45#22548635C2211yescharacteristic, complemented
46#22548635C2211yescharacteristic, complemented
47C22⋊(C251⋊C8)C2211yescharacteristic, complemented
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49#22548626C2211yescharacteristic, complemented
50C2×#41×C251C2211yescharacteristic, complemented
51C22×Q8×C251C2211yescharacteristic, complemented
52#124×C251—21no—
53#124×C251C2211yescharacteristic, complemented
54C22⋊Q8×C251—21no—
55C22⋊Q8×C251C2211yescharacteristic, complemented
56#121×C251—21no—
57#121×C251C2211yescharacteristic, complemented
58C4×Q8×C251C2211yescharacteristic, complemented
59C4×D4×C251—21no—
60C4×D4×C251C2211yescharacteristic
61C23×D251—21no—
62C23×D251—21no—
63C2×D1004—22no—
64C2×D1004—22no—
65C2×D1004—21no—
66C2×D1004—21no—
67C2×D1004—21no—
68C2×D1004—21no—
69C2×D1004—41no—
70C2×D1004—41no—
71C2×D1004C2311yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76D4×D251—44no—
77C2×C4×D251—21no—
78#16860797—22no—
79#16860797—22no—
80#16860797—22no—
81#16860797—22no—
82#16860797—41no—
83#16860797D414yescomplemented
84#16860795—22no—
85#16860795—22no—
86#16860795—41no—
87#16860794—22no—
88C2×C251⋊C8—22no—
89C2×C251⋊C8C2311yescharacteristic
90C2×C251⋊C8C2311yescharacteristic
91C2×C251⋊C8C2311yescharacteristic
92C2×C251⋊C8C2311yescharacteristic
93C4⋊Dic251C2311yescharacteristic
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251—21no—
100C2×Q8×C251—22no—
101C2×Q8×C251—41no—
102C2×Q8×C251C2311yescharacteristic
103C2×Q8×C251C2311yescharacteristic
104C2×D4×C251—21no—
105C2×D4×C251—21no—
106C2×D4×C251D411yescharacteristic
107C2×D4×C251C2311yescharacteristic
108C22×C4×C251—21no—
109C22×C4×C251—21no—
110C22×C4×C251C2311yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C4⋊C4×C251—21no—
113C4⋊C4×C251—21no—
114C4⋊C4×C251D411yescharacteristic
115C4⋊C4×C251C2311yescharacteristic
116C22⋊C4×C251—21no—
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C22⋊C4×C251—21no—
120C22⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C42×C251—21no—
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—81no—
132C22×D251—81no—
133D1004—21no—
134D1004—21no—
135D1004—21no—
136D1004—21no—
137D1004—21no—
138D1004—42no—
139D1004—42no—
140D1004—42no—
141D1004—81no—
142D1004C2×D412yes—
143C22⋊D251—42no—
144C22⋊D251—42no—
145C2×Dic251—21no—
146C4×D251—42no—
147C251⋊C8—21no—
148C251⋊C8—21no—
149C251⋊C8—21no—
150C251⋊C8C2×D412yescomplemented
151C23×C251—21no—
152C23×C251C2×D411yescharacteristic
153C23×C251C2×D411yescharacteristic
154Q8×C251—21no—
155Q8×C251—21no—
156Q8×C251—21no—
157Q8×C251—42no—
158Q8×C251C2×D412yescomplemented
159D4×C251—22no—
160D4×C251—21no—
161D4×C251—21no—
162D4×C251—42no—
163D4×C251—42no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—42no—
172C2×C4×C251—42no—
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2×D411yescharacteristic
176C2×C4×C251C2×D411yescharacteristic
177C2×C4×C251C2411yescharacteristic
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—21no—
181C22×C251—21no—
182C22×C251—21no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251AGL(1,ℤ/8)12yescomplemented
187C22×C251C22×D411yescharacteristic
188C2×D251—22no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—84no—
198C2×D251—84no—
199C4×C251—21no—
200C4×C251—21no—
201C4×C251—22no—
202C4×C251—21no—
203C4×C251—21no—
204C4×C251—41no—
205C4×C251—41no—
206C4×C251C22×D411yescharacteristic
207C4×C251H(5,𝔽2)11yescharacteristic
208Dic251—41no—
209C2×C251—22no—
210C2×C251—41no—
211C2×C251—81no—
212C2×C251D4211yescharacteristic
213C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C251#57711yescharacteristic
215D251—42no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548D25111yescharacteristic, complemented
223D42—2511no—
224#496—2511no—
225#468—2511no—
226#466—2511no—
227#463—2511no—
228#460—2511no—
229#450—2511no—
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435—2511no—
235C2×#41C2×D25111yescharacteristic, complemented
236C22×Q8C2×D25111yescharacteristic, complemented
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4—2511no—
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16—2511no—
247C2×SD16—2511no—
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16—2511no—
252C22×C8—2511no—
253#127—2511no—
254#124—21no—
255#124C2×D25111yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D25111yescharacteristic, complemented
258#121—21no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121C2×D25111yescharacteristic, complemented
263#120—5021no—
264#120—5021no—
265C4×Q8C2×D25111yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—2511no—
268C4×D4C2×D25111yescharacteristic
269#107—2511no—
270C4⋊C8—2511no—
271#103—2511no—
272#103—2511no—
273#102—2511no—
274#102—2511no—
275#102—2511no—
276#102—2511no—
277C22⋊C8—2511no—
278C22⋊C8—2511no—
279#97—2511no—
280C24—5021no—
281C24—5021no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D25111yescharacteristic
288C2×Q8C22×D25111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—10041no—
292C2×D4—10041no—
293C2×D4—10044no—
294C2×D4—5022no—
295C2×D4—5022no—
296C2×D4—5021no—
297C2×D4—5021no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4C22⋊D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—5021no—
310C22×C4C22×D25111yescharacteristic
311C22×C4C22×D25111yescharacteristic
312SD16—10041no—
313SD16—5022no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—5022no—
322C2×C8—5022no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8—2511no—
326C2×C8—2511no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—2511no—
330C4⋊C4C22⋊D25111yescharacteristic
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4C22⋊D25111yescharacteristic
340C22⋊C4C22⋊D25111yescharacteristic
341C42—21no—
342C42C22×D25111yescharacteristic
343C23—21no—
344C23—20081no—
345C23—20081no—
346C23—10042no—
347C23—10042no—
348C23—10042no—
349C23—10042no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23C2×C22⋊D25111yescharacteristic
354C23C2×C22⋊D25111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D25112yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—20081no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—10042no—
370D4—10042no—
371D4—5021no—
372D4—5021no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4—2512no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—10042no—
387C2×C4—5021no—
388C2×C4C2×C22⋊D25111yescharacteristic
389C2×C4C2×C22⋊D25111yescharacteristic
390C2×C4C2×C22⋊D25111yescharacteristic
391C2×C4C2×C22⋊D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—5021no—
394C8—5021no—
395C8—5021no—
396C8—2512no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—20084no—
406C22—20084no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—10041no—
414C22—10041no—
415C22—5022no—
416C22#2254874012yescomplemented
417C22C22×C22⋊D25111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—10041no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876011yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—20081no—
432C2—20081no—
433C2—10042no—
434C2D4×C22⋊D25111yescharacteristic
435C2C2×#2254874011yescharacteristic
436C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#548×C251C2
socleC22×C251C22×D4
hypercenter#548D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C22×#16860797C2semidirect
3. D4×D1004C2semidirect
4. #30632077C2semidirect
5. #30631935C2semidirect
6. #30631921C2semidirect
7. #30631899C2semidirect
8. #30631887C2semidirect
9. #30631837C2semidirect
10. #30631832C2semidirect
11. #30631826C2semidirect
12. #30631820C2semidirect
13. #30631785C2semidirect
14. #30631759C2semidirect
15. #548×C251C2semidirect
23. C2×#16860797C22semidirect
24. C2×#16860797C22semidirect
25. C2×#16860797C22semidirect
26. C2×#16860797C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860795C22semidirect
30. C2×#16860794C22semidirect
31. C22×C251⋊C8C22semidirect
39. #22548648C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
43. #22548636C22semidirect
45. #22548635C22semidirect
46. #22548635C22semidirect
47. C22⋊(C251⋊C8)C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. #22548626C22semidirect
50. C2×#41×C251C22semidirect
51. C22×Q8×C251C22semidirect
53. #124×C251C22semidirect
55. C22⋊Q8×C251C22semidirect
57. #121×C251C22semidirect
58. C4×Q8×C251C22semidirect
83. #16860797D4semidirect
150. C251⋊C8C2×D4semidirect
158. Q8×C251C2×D4semidirect
186. C22×C251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
222. #548D251semidirect
235. C2×#41C2×D251semidirect
236. C22×Q8C2×D251semidirect
255. #124C2×D251semidirect
257. C22⋊Q8C2×D251semidirect
262. #121C2×D251semidirect
265. C4×Q8C2×D251semidirect
359. Q8D4×D251semidirect
416. C22#22548740semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225487402
C22×#168607972
D4×D10042
#306320772
#306319352
#306319212
#306318992
#306318872
#306318372
#306318322
#306318262
#306318202
#306317852
#306317592
#548×C2512
#3265251